First Order Linear Differential Equations and Largest Interval over which the solution is defined

Introduction

In this tutorial, we will explore how to solve first-order linear differential equations and identify the largest interval over which the solution is defined. Understanding these concepts is crucial for advanced studies in differential equations, and this guide provides a structured approach to mastering them.

Step 1: Understanding First Order Linear Differential Equations

First-order linear differential equations have the standard form:

[ \frac{dy}{dx} + P(x)y = Q(x) ]

Where:

• ( P(x) ) and ( Q(x) ) are continuous functions of ( x ).
• ( y ) is the dependent variable.

Practical Advice

• Ensure that ( P(x) ) and ( Q(x) ) are continuous on the interval you are considering. This continuity is essential for applying the methods discussed in this tutorial.

Step 2: Finding the Integrating Factor

To solve the equation, we need to find an integrating factor, ( \mu(x) ), defined as:

[ \mu(x) = e^{\int P(x) , dx} ]

Steps to Calculate the Integrating Factor

1. Calculate the integral of ( P(x) ).
2. Exponentiate the result to find ( \mu(x) ).

Example

If ( P(x) = 2x ):

• Calculate ( \int 2x , dx = x^2 ).
• Thus, ( \mu(x) = e^{x^2} ).

Step 3: Multiplying the Equation by the Integrating Factor

Multiply the entire differential equation by ( \mu(x) ):

[ \mu(x) \left( \frac{dy}{dx} + P(x)y \right) = \mu(x)Q(x) ]

Practical Advice

• This step transforms the left side into a derivative of the product ( \mu(x)y ).

Step 4: Integrating Both Sides

Now, integrate both sides with respect to ( x ):

[ \int \frac{d(\mu(x)y)}{dx} , dx = \int \mu(x)Q(x) , dx ]

Practical Advice

• This will yield the solution in the form of ( \mu(x)y = \int \mu(x)Q(x) , dx + C ), where ( C ) is the constant of integration.

Step 5: Solving for ( y )

Isolate ( y ) to find the general solution:

[ y = \frac{1}{\mu(x)} \left( \int \mu(x)Q(x) , dx + C \right) ]

Step 6: Determining the Largest Interval of Definition

To find the largest interval over which the solution is defined:

1. Identify where the functions ( P(x) ) and ( Q(x) ) are continuous.
2. The largest interval is determined by the points where these functions are undefined.

Practical Tip

• Look for discontinuities in ( P(x) ) and ( Q(x) ) to establish boundaries for the solution's interval.

Conclusion

You have now learned how to solve first-order linear differential equations and determine the largest interval for solutions. Remember to:

• Check the continuity of ( P(x) ) and ( Q(x) ).
• Calculate the integrating factor accurately.
• Integrate both sides carefully to find the general solution.

For further practice, explore different forms of ( P(x) ) and ( Q(x) ) to solidify your understanding of these concepts.